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Register a new user$lambda$-analogues of r-Stirling numbers of the first kind
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In this paper, we study $lambda$-analogues of the r-Stirling numbers of the first kind which have close connections with the r-Stirling numbers of the first kind and $lambda$-Stirling numbers of the first kind. Specifically, we give the recurrence relations for these numbers and show their connections with the $lambda$-Stirling numbers of the first kind and higher-order Daehee polynomials.
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Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_2(s(2^n,k))$ holds for all integers $k$ between 1 and $2^n$. As a corollary, we show that $v_2(s(2^n,2^n-k))=2n-2-v_2(k-1)$ if $k$ is odd and $2le kle 2^{n-1}+1$. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if $kle 2^n$, then $v_2(s(2^n,k)) le v_2(s(2^n,1))$ and $v_2(H(2^n,k))leq -n$, where $H(n,k)$ stands for the $k$-th elementary symmetric functions of $1,1/2,...,1/n$. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.
Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washingtons congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound of $v_p(s(ap, k))$ with $a$ and $k$ being integers such that $1le ale p-1$ and $1le kle ap$. This infers that for any regular prime $pge 7$ and for arbitrary integers $a$ and $k$ with $5le ale p-1$ and $a-2le kle ap-1$, one has $v_p(H(ap-1,k))<-frac{log{(ap-1)}}{2log p}$ with $H(ap-1, k)$ being the $k$-th elementary symmetric function of $1, frac{1}{2}, ..., frac{1}{ap-1}$. This gives a partial support to a conjecture of Leonetti and Sanna raised in 2017. We also present results on $v_p(s(ap^n,ap^n-k))$ from which one can derive that under certain condition, for any prime $pge 5$, any odd number $kge 3$ and any sufficiently large integer $n$, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-))=v_p(s(ap^n,ap^n-k))+2$. It confirms partially Lengyels conjecture proposed in 2015.
In this paper, we consider the degenerate Changhee numbers and polynomials of the second kind which are different from the previously introduced degenerate Changhee numbers and polynomials by Kwon-Kim-Seo (see [11]). We investigate some interesting identities and properties for these numbers and polynomials. In addition, we give some new relations between the degenerate Changhee polynomials of the second kind and the Carlitzs degenerate Euler polynomials.
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating function for those numbers. We obtain explicit expressions for the coefficients appearing in those differential equations and the degenerate Bernoulli numbers of the second kind. In addition, as an application and from those differential equations we have an identity expressing the degenerate Bernoulli numbers of the second kind in terms of those numbers of higher-orders.
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)frac{(frac{1}{2})_k^3}{k!^3} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ sum_{k=0}^{p-1}(-1)^k (3k+1)frac{(frac{1}{2})_k^3}{k!^3} 2^{3k} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supercongruences by employing the $q$-WZ method. As a conclusion, we provide a $q$-analogue of the following supercongruence of Sun: $$ sum_{k=0}^{(p-1)/2}frac{(frac{1}{2})_k^2}{k!^2} equiv (-1)^{(p-1)/2}+p^2 E_{p-3} pmod{p^3}. $$
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